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    %% Evaluate Double Integral in Polar Coordinates  

%% 
% Define the function
%
% $$f \left( \theta,r \right) = \frac{r}{\sqrt{r \cos \theta + r \sin
% \theta} \left( 1+r \cos \theta + r \sin \theta \right) ^2}$$
%
fun = @(x,y) 1./( sqrt(x + y) .* (1 + x + y).^2 );
polarfun = @(theta,r) fun(r.*cos(theta),r.*sin(theta)).*r;  

%% 
% Define a function for the upper limit of $r$. 
rmax = @(theta) 1./(sin(theta) + cos(theta));  

%% 
% Integrate over the region bounded by $0 \le \theta \le \pi/2$ and $0 \le
% r \le r_{max}$.
q = integral2(polarfun,0,pi/2,0,rmax)